Let (**K**
_{α}: *α* ≧ *ω*) be a system of varieties definable by schemas. We characterize the amalgamation base, strong amalgamation base, and super amalgamation base of the class **K**
_{α+ω} at this abstract level.

We construct an infinite dimensional quasi-polyadic equality algebra

Let *α* be an infinite ordinal. Let RCA_{α} denote the variety of representable cylindric algebras of dimension *α*. Modifying Andréka’s methods of splitting, we show that the variety RQEA_{α} of representable quasi-polyadic equality algebras of dimension *α* is not axiomatized by a set of universal formulas containing only finitely many variables over the variety RQA_{α} of representable quasi-polyadic algebras of dimension *α*. This strengthens a seminal result due to Sain and Thompson, answers a question posed by Andréka, and lifts to the transfinite a result of hers proved for finite dimensions > 2. Using the modified method of splitting, we show that all known complexity results on universal axiomatizations of RCA_{α} (proved by Andréka) transfer to universal axiomatizations of RQEA_{α}. From such results it can be inferred that any algebraizable extension of *L*_{ω,ω} is severely incomplete if we insist on Tarskian square semantics. Ways of circumventing the strong non-negative axiomatizability results hitherto obtained in the first part of the paper, such as guarding semantics, and /or expanding the signature of RQEA_{ω} by substitutions indexed by transformations coming from a finitely presented subsemigroup of (^{ω}*ω*, ○) containing all transpositions and replacements, are surveyed, discussed, and elaborated upon.

## Abstract

Fix 2 < n < ω and let CA_{n} denote the class of cyindric algebras of dimension n. Roughly CA_{n} is the algebraic counterpart of the proof theory of first order logic restricted to the first *n* variables which we denote by L_{n}. The variety RCA_{n} of representable CA_{n}s reflects algebraically the semantics of *L*
_{n}. Members of RCA_{n} are concrete algebras consisting of genuine n-ary relations, with set theoretic operations induced by the nature of relations, such as projections referred to as cylindrifications. Although CA_{n} has a finite equational axiomatization, RCA_{n} is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CA_{n} substantially richer than that of Boolean algebras, just as much as L_{ω,ω} is richer than propositional logic. We show using a so-called blow up and blur construction that several varieties (in fact infinitely many) containing and including the variety RCA_{n} are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever **𝔄**
*L*
_{ω,ω}, fails dramatically for *L*
_{n} even if we allow certain generalized models that are only locallly clasfsical. It is also shown that any class K such that _{n} is the class of completely representable CA_{n}s, and S_{c} denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that _{d} denotes the operation of forming dense subalgebra.